reserve X for set,
        A for Subset of X,
        R,S for Relation of X;
reserve QUS for Quasi-UniformSpace;
reserve SUS for Semi-UniformSpace;
reserve T for TopSpace;

theorem
  for T being non empty strict TopSpace holds
    TopSpace_induced_by(Pervin_quasi_uniformity T) = T
  proof
    let T be non empty strict TopSpace;
    the topology of TopSpace_induced_by(Pervin_quasi_uniformity T)
      = Family_open_set(FMT_induced_by(Pervin_quasi_uniformity T))
      by FINTOPO7:def 16;
    hence thesis by FINTOPO7:def 16,Th32;
  end;
